Editing Lattice (group) (section)

==Lattices in general vector spaces==
{{unreferenced section|date=April 2022}}
While we normally consider <math>\mathbb{Z}</math> lattices in <math>\mathbb{R}^n</math> this concept can be generalized to any finite-dimensional [[vector space]] over any [[Field (mathematics)|field]]. This can be done as follows:

Let ''K'' be a [[field (mathematics)|field]], let ''V'' be an ''n''-dimensional ''K''-[[vector space]], let <math>B = \{\mathbf{v}_1,\ldots, \mathbf{v}_n\}</math> be a ''K''-[[basis (linear algebra)|basis]] for ''V'' and let ''R'' be a [[Ring (mathematics)|ring]] contained within ''K''. Then the ''R'' lattice <math>\mathcal{L}</math> in ''V'' generated by ''B'' is given by:

:<math>\mathcal{L} = \biggl\{ \sum_{i=1}^{n} a_i \mathbf{v}_i \mathbin{\bigg\vert} a_i \in R\biggr\}.</math>

In general, different bases ''B'' will generate different lattices. However, if the [[Change of basis#General case|transition matrix]] ''T'' between the bases is in <math>\mathrm{GL}_n(R)</math> - the [[general linear group]] of ''R'' (in simple terms this means that all the entries of ''T'' are in ''R'' and all the entries of <math>T^{-1}</math> are in ''R'' - which is equivalent to saying that the [[determinant]] of ''T'' is in <math>R^*</math> - the [[unit group]] of elements in ''R'' with multiplicative inverses) then the lattices generated by these bases will be [[isomorphism|isomorphic]] since ''T'' induces an isomorphism between the two lattices.

Important cases of such lattices occur in number theory with ''K'' a [[p-adic field|''p''-adic field]] and ''R'' the [[p-adic integers|''p''-adic integer]]s.

For a vector space which is also an [[inner product space]], the [[dual lattice]] can be concretely described by the set

:<math>\mathcal{L}^* = \{\mathbf{v} \in V \mid \langle \mathbf{v},\mathbf{x} \rangle \in R\,\text{ for all }\,\mathbf{x} \in \mathcal{L}\},</math>

or equivalently as

:<math>\mathcal{L}^* = \{\mathbf{v} \in V \mid \langle \mathbf{v},\mathbf{v}_i \rangle \in R,\ i = 1, ..., n\}.</math>